r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N651-B:  N = 651;  Sigma* = 1/2 (computed);  nu = 102451/12108600000, x0 = 1/5000, X = 500631032276116980167182393677011/500000000000000000000000000000000, rho = 899032669057340117937713848273401/100000000000000000000000000000000000000
  [1] (2), (3), (4) vanish exactly at the six states of Lemma 3.2, all C_j > 0, Sigma = Sigma*, feasible: True;  signs of the right-hand side of (1): + - + - + -   PASS
  [2] signs of P at the six test points: + - + - + -   PASS   (integers chat_1, that_0 of 21756 to 83876 bits)
      time: 1.9 s
  [3] Lemma 4.1 (PB = 320): 104422 closed intervals cover [x0, X]: 104400 without a root, 22 monotone; deepest bisection 39, 208818 point evaluations, smallest interval 1.82e-12
      roots of P in [x0, X]: 5, all simple;  signs of P': - + - + -   PASS   (one in each interval between consecutive test points: True)
        root 1:  x in (0.9849720038727526, 0.9849720038763945),  S in (0.9847720038727527, 0.9847720038763945)
        root 2:  x in (0.9850703921885805, 0.9850703921904014),  S in (0.9848703921885805, 0.9848703921904014)
        root 3:  x in (0.9852197497919919, 0.9852197497938128),  S in (0.9850197497919919, 0.9850197497938128)
        root 4:  x in (0.9854192518092854, 0.9854192518111062),  S in (0.9852192518092854, 0.9852192518111063)
        root 5:  x in (0.9855059684056943, 0.9855059684093361),  S in (0.9853059684056943, 0.9853059684093360)
      time: 230.1 s
  [4] coefficients of a positive multiple of P: degree 652, up to 21944 bits   (time: 1.0 s)
      Horner values of the coefficient list at the six test points equal the values of the recursion: True
      back end python-flint 0.9.0: 13 intervals between x0 and X, sign changes: 0 0 0 0 0 1 1 1 1 1 0 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 197/200, 2463/2500, 4927/5000, 1971/2000, 493/500, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 5.5 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 13 intervals between x0 and X, sign changes: 0 0 0 0 0 1 1 1 1 1 0 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 197/200, 2463/2500, 4927/5000, 1971/2000, 493/500, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 11.0 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
  [5] gcd(P, P') is constant (all complex roots of P are simple): True   PASS   (time: 0.0 s)
  N651-B: ALL EXECUTED PARTS PASS   (total 250.7 s)
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1 parameter points: ALL PASS
